106 XIII INTEGRATION elsewhere (/= 1, 2). Then /?f is continuous on X, because /f/(/i +/2) is continuous at the points x where f^x) Jrf2(x) > 0, and also |/Zj(x)| <£ |/?(x)| for all jc G X, which proves that ht is continuous at the points x where /iOO +/2(X)= 0, because /?(*) also vanishes at these points. It is clear that | A, | <*/i (z = 1, 2) and that h = hi + h2; hence Since |ju(/?)| may be taken to be arbitrarily close to pC/i + /2), it follows that P(/i + /2) £ P(/I) + p(/2). Hence (13.3.2.2) is proved. We shall now extend the definition of />(/) to all functions/e JfR(X). To do this we write p(/) = p(f) - p(/")> where/=/' — f" is any decomposition of /as the difference of two functions/',/" ^ 0 belonging to JfR(X). The value of p(/) so obtained is independent of the decomposition, because if /=/i -/? =/2 -/!'> then // +/J =/^' +/2' and therefore p(f[) + p(/^) = P(/D + P(/2> by virtue of (13.3.12). With this definition, the formula (13.3.2.2) is valid for all/j.,/2 in JfR(X). For we can write/! =// -f'^f2 =f2 -f£ where// 9f? (i = 1, 2) are ^0 and belong to Jf R(X); since /x +/2 = (// +/2f) - t/i" +/20» our assertion follows from the definition above and from (13.3.2.2) for functions ^0. Finally, the above definition shows that, for each scalar a ^ 0, we have p(af) = ap(f); and if a < 0, then we have p(af) = p(af - of) = p(-aD + p(af) Hence the relation p(tz/) = ap(f) is valid for all real scalars a, and therefore we have proved that p is a (real) linear form on JTR(X). Hence by (13.3.1) it is a positive measure. Q.E.D. The measure p defined by (13.3.2) is called the absolute value of the complex measure //, and is denoted by |/*|. Hence by definition we have (133.3) |M/)| £ l/ild/l) for all /e Jfc(X). It is immediately seen that if a e C and p is a measure on X, we have (13.3.4) M = H'M. If i*. is a positive measure on X, then (13.3.5) |,i| = /i.measure on Ua such that, for each pair of indices a, j8, the restrictions of i*a